Integration - Building Totals from Rates of Change
Integration is the reverse of differentiation. If differentiation breaks things apart to find rates of change, integration builds them back together to find totals – areas, volumes, accumulated quantities. Integration is one of the most powerful tools in all of mathematics and science.
Two Types of Integration
Indefinite integrals find a general family of functions (always including + C).
Definite integrals calculate a specific numerical value – the net area between a curve and the x-axis over a given interval.
The Antiderivative
The integral of f(x) is a function F(x) such that F′(x) = f(x). It is called the antiderivative.
Notation: ∫ f(x) dx = F(x) + C
The constant C appears because many functions share the same derivative – for example, x², x²+5, and x²−100 all differentiate to 2x.
Basic Integration Rules
| Function f(x) | Integral ∫f(x) dx |
|---|---|
| xn (n ≠ −1) | xn+1 / (n+1) + C |
| 1/x | ln|x| + C |
| ex | ex + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| sec² x | tan x + C |
| k (constant) | kx + C |
The Fundamental Theorem of Calculus
This theorem is the bridge between differentiation and integration. It has two parts:
Part 1: If F(x) = ∫ax f(t) dt, then F′(x) = f(x). Differentiation undoes integration.
Part 2: ∫ab f(x) dx = F(b) − F(a), where F is any antiderivative of f.
Part 2 is the practical rule you use to evaluate definite integrals.
Worked Examples
Integrate term by term:
∫ 3x² dx = x³, ∫ 4x dx = 2x², ∫ −5 dx = −5x.
Answer: x³ + 2x² − 5x + C.
Antiderivative: F(x) = x² + x.
F(3) − F(1) = (9 + 3) − (1 + 1) = 12 − 2 = 10.
∫03 x² dx = [x³/3]03 = 27/3 − 0 = 9 square units.
Use reverse chain rule. The antiderivative of sin(u) is −cos(u); the inner derivative of 2x is 2.
∫ sin(2x) dx = −cos(2x)/2 + C.
Integration by Substitution
When an integrand contains a composite function, substitute u = g(x) to simplify it.
Steps: 1. Choose u = inner function. 2. Find du/dx and write dx = du / g′(x). 3. Rewrite the integral in terms of u. 4. Integrate. 5. Back-substitute x.
Let u = x² + 1. Then du = 2x dx.
∫ u4 du = u5/5 + C = (x²+1)5/5 + C.
Key Takeaways
- Integration is the reverse of differentiation.
- Power rule: ∫ xn dx = xn+1/(n+1) + C (n ≠ −1).
- Definite integral = F(b) − F(a) = exact area under the curve from a to b.
- Always add + C to indefinite integrals.
Practice Questions
- Find ∫ (5x4 − 6x + 3) dx.
- Evaluate ∫02 (x² + 2) dx.
- Find the area under y = 3x² between x = 1 and x = 4.
- Find ∫ cos(4x) dx.
- Use substitution to find ∫ 3x²(x³ + 5)6 dx.